For a graph G without isolated vertices, the inverse degree of a graph G is defined as ID(G)=∑u∈V(G)d(u)−1
Nearby in the stack
where
d(u)
is the number of vertices adjacent to the vertex
u
in
G
. By replacing
−1
by any non-zero real number we obtain zeroth-order general Randić index, i.e.
0Rγ(G)=∑u∈V(G)d(u)γ
where
γ
is any non-zero real number. In xd, Xu et. al. determined some upper and lower bounds on the inverse degree for a connected graph
G
in terms of chromatic number, clique number, connectivity, number of cut edges. In this paper, we extend their results and investigate if the same results hold for
γ<0
. The corresponding extremal graphs have been also characterized.